Seminars and Colloquia by Series

Intrinsic Combinatorics for the Space of Generic Complex Polynomials

Series
Geometry Topology Seminar
Time
Monday, September 23, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Michael Dougherty – Colby College –

The space of degree-n complex polynomials with distinct roots appears frequently and naturally throughout mathematics, but its shape and structure could be better understood. In recent and ongoing joint work with Jon McCammond, we present a deformation retraction of this space onto a simplicial complex with rich structure given by the combinatorics of noncrossing partitions. In this talk, I will describe the deformation retraction and the resulting combinatorial data associated to each generic complex polynomial. We will also discuss a helpful comment from Daan Krammer which connects our work with the ideas of Bill Thurston and his collaborators.

The “generating function” of configuration spaces, as a source for explicit formulas and representation stability

Series
Geometry Topology Seminar
Time
Monday, September 16, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Nir Gadish – Massachusetts Institute of Technology –

As countless examples show, sequences of complicated objects should be studied all at once via the formalism of generating functions. We apply this point of view to the homology and combinatorics of (orbit-)configuration spaces: using the notion of twisted commutative algebras, which categorify exponential generating functions. With this idea the configuration space “generating function” factors into an infinite product, whose terms are surprisingly easy to understand. Beyond the intrinsic aesthetic of this decomposition and its quantitative consequences, it also gives rise to representation stability - a notion of homological stability for sequences of representations of differing groups.

Link Concordance and Groups

Series
Geometry Topology Seminar
Time
Monday, September 9, 2019 - 14:00 for
Location
Speaker
Miriam Kuzbary – Georgia Tech –

This is a general audience Geometry-Topology talk where I will give a broad overview of my research interests and techniques I use in my work.  My research concerns the study of link concordance using groups, both extracting concordance data from group theoretic invariants and determining the properties of group structures on links modulo concordance. Milnor's invariants are one of the more fundamental link concordance invariants; they are thought of as higher order linking numbers and can be computed using both Massey products (due to Turaev and Porter) and higher order intersections (due to Cochran). In my work, I have generalized Milnor's invariants to knots inside a closed, oriented 3-manifold M. I call this the Dwyer number of a knot and show methods to compute it for null-homologous knots inside a family of 3-manifolds with free fundamental group. I further show Dwyer number provides the weight of the first non-vanishing Massey product in the knot complement in the ambient manifold. Additionally, I proved the Dwyer number detects knots K in M bounding smoothly embedded disks in specific 4-manifolds with boundary M which are not concordant to the unknot in M x I. This result further motivates my definition of a new link concordance group in joint work with Matthew Hedden using the knotification construction of Ozsv'ath and Szab'o. Finally, I will briefly discuss my recent result that the string link concordance group modulo its pure braid subgroup is non-abelian.

Dynamical Mapping Classes

Series
Geometry Topology Seminar
Time
Monday, August 26, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jasmine Powell – University of Michigan –

In complex dynamics, the main objects of study are rational maps on the Riemann sphere. For some large subset of such maps, there is a way to associate to each map a marked torus. Moving around in the space of these maps, we can then track the associated tori and get induced mapping classes. In this talk, we will explore what sorts of mapping classes arise in this way and use this to say something about the topology of the original space of maps.

Group Actions and Cogroup Coactions in Simplicial Sheaves

Series
Geometry Topology Seminar
Time
Tuesday, August 13, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skile 114
Speaker
Jonathan Beardsley – Georgia Tech

Please Note: Please note different day and room.

In this talk, I will describe joint work with Maximilien Péroux on understanding Koszul duality in ∞-topoi. An ∞-topos is a particularly well behaved higher category that behaves like the category of compactly generated spaces. Particularly interesting examples of ∞-topoi are categories of simplicial sheaves on Grothendieck topologies. The main theorem of this work is that given a group object G of an ∞-topos, there is an equivalence of categories between the category of G-modules in that topos and the category of BG-comodules, where BG is the classifying object for G-torsors. In particular, given any pointed space X, the space of loops on X, denoted ΩX, can be lifted to a group object of any ∞-topos, so if X is in addition a connected space, there is an equivalence between objects of any ∞-topos with an ΩX-action, and objects with an X-coaction (where X is a coalgebra via the usual diagonal map). This is a generalization of the classical equivalence between G-spaces and spaces over BG for G a topological group.

Factorization homology: sigma-models as state-sum TQFTs.

Series
Geometry Topology Seminar
Time
Friday, May 31, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
David Ayala – Montana State University
Roughly, factorization homology pairs an n-category and an n-manifold to produce a vector space.  Factorization homology is to state-sum TQFTs as singular homology is to simplicial homology: the former is manifestly well-defined (ie, independent of auxiliary choices), continuous (ie, beholds a continuous action of diffeomorphisms), and functorial; the latter is easier to compute.  
 
Examples of n-categories to input into this pairing arise, through deformation theory, from perturbative sigma-models.  For such n-categories, this state sum expression agrees with the observables of the sigma-model — this is a form of Poincare’ duality, which yields some surprising dualities among TQFTs.  A host of familiar TQFTs are instances of factorization homology; many others are speculatively so.  
 
The first part of this talk will tour through some essential definitions in what’s described above.  The second part of the talk will focus on familiar manifold invariants, such as the Jones polynomial, as instances of factorization homology, highlighting the Poincare’/Koszul duality result.  The last part of the talk will speculate on more such instances.  

Rational cobordisms and integral homology

Series
Geometry Topology Seminar
Time
Wednesday, May 29, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Paolo Aceto – University of Oxford –

We prove that every rational homology cobordism class in the subgroup generated by lens spaces contains a unique connected sum of lens spaces whose first homology embeds in any other element in the same class. As a consequence we show that several natural maps to the rational homology cobordism group have infinite rank cokernels, and obtain a divisibility condition between the determinants of certain 2-bridge knots and other knots in the same concordance class. This is joint work with Daniele Celoria and JungHwan Park.

Flag moduli spaces and Legendrian surfaces

Series
Geometry Topology Seminar
Time
Wednesday, May 15, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skile 005
Speaker
Roger Casals – UC Davis

In this talk, I will discuss progress in our understanding of Legendrian surfaces. First, I will present a new construction of Legendrian surfaces and a direct description for their moduli space of microlocal sheaves. This Legendrian invariant will connect to classical incidence problems in algebraic geometry and the study of flag varieties, which we will study in detail. There will be several examples during the talk and, in the end, I will indicate the relation of this theory to the study of framed local systems on a surface. This talk is based on my work with E. Zaslow.

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